analysis

Throughout this article are references to a variety of
number systems—that is, collections of mathematical objects (numbers) that can be operated on by some or all of the standard operations of arithmetic: addition, multiplication, subtraction, and division. Such systems have a variety of technical names (e.g., group, ring, field) that are not employed here. This article shall, however,...

ancient Middle East

...both deities and personified numbers. The planet Venus was the “star” that the Assyrians and Babylonians called Ishtar, which was at the same time both the goddess Ishtar and the deified number 15. The Moon was not only Earth’s satellite but also the lunar deity Sin and the deified number 30. The most perfect number was one, for by advancing from zero to one men believed they...

characteristics of

Austronesian languages

Most Austronesian languages have a decimal system of counting. Others, such as Ilongot of the northern Philippines and some of the languages of the Lesser Sunda Islands in eastern Indonesia, have quinary systems (i.e., systems based on five). In the New Guinea area several Austronesian languages have radically restructured
number systems...

contribution by Cantor

...mathematician Bernhard Riemann (
q.v.) in 1854, Cantor in 1870 showed that such a function can be represented in only one way by a trigonometric series. Consideration of the collection of numbers (points) that would not conflict with such a representation led him, first, in 1872, to define irrational numbers in terms of convergent sequences of rational numbers (quotients of integers)...

foundations of mathematics

While the ancient Greeks were familiar with the positive integers, rationals, and reals, zero (used as an actual number instead of denoting a missing number) and the negative numbers were first used in India, as far as is known, by Brahmagupta in the 7th century
ce. Complex numbers were introduced by the Italian Renaissance mathematician and physician Gerolamo Cardano (1501–76), not...

...and coherent conception of the notion of equation that served as a broadly accepted starting point for later developments. No similar single reference point exists for the general conception of number, however. Some significant milestones may nevertheless be mentioned, and prominent among them was
De Thiende (
Disme: The Art of Tenths), an...

history of science

...The problem of form was first attacked systematically by the philosopher and cult leader Pythagoras in the 6th century
bce. Legend has it that Pythagoras became convinced of the primacy of number when he realized that the musical notes produced by a monochord were in simple ratio to the length of the string. Qualities (tones) were reduced to quantities (numbers in integral ratios)....

philosophical considerations

Pre-Socratics

All of the philosophies mentioned so far are in various ways historically akin to one another. Toward the end of the 6th century
bc, however, there arose, quite independently, another kind of philosophy, which only later entered into interrelation with the developments just mentioned: the philosophy of Pythagoras of Samos (
c. 580–
c. 500
bc

Pythagoreanism

...Pythagoreans invariably supported their doctrines by indiscriminately citing their master’s authority. Pythagoras, however, is generally credited with the theory of the functional significance of numbers in the objective world and in music. Other discoveries often attributed to him (the incommensurability of the side and diagonal of a square, for example, and the Pythagorean theorem for right...

...about the stuff out of which things were ultimately made, but a new twist was given to the inquiry when Pythagoras, in the late 6th century
bc, arrived at the answer that what was really there was number. Pythagoras conceived what is there in terms not of matter but of intelligible structure; it was the latter that gave each type of thing its distinctive character and made it what it was. The...

...confusing. Its fame rests, however, on some very influential ideas, not always correctly understood, that have been ascribed to it since antiquity. These ideas include those of (1) the metaphysic of number and the conception that reality, including music and astronomy, is, at its deepest level, mathematical in nature; (2) the use of philosophy as a means of spiritual purification; (3) the...

...of opposites, and whatever reflections of Eastern mathematics there are in Pythagoreanism; and from the technicians of his birthplace, the Isle of Samos, he learned to understand the importance of number, measurements, and proportions. Popular cults and beliefs current in the 6th century and reflected in the tenets of Orphism introduced him to the notions of occultism and ritualism and to the...

rationalist epistemology

...relation to the lengths of the strings, Pythagoras held that these harmonies reflected the ultimate nature of reality. He summed up the implied metaphysical rationalism in the words “All is number.” It is probable that he had caught the rationalist’s vision, later seen by Galileo (1564–1642), of a world governed throughout by mathematically formulable laws.

physical sciences

Modern physical science is characteristically concerned with numbers—the measurement of quantities and the discovery of the exact relationship between different measurements. Yet this activity would be no more than the compiling of a catalog of facts unless an underlying recognition of uniformities and correlations enabled the investigator to choose what to measure out of an infinite...

religious symbolism

...the power of the spirit are concretely expressed in religious terms. The idea of unity plays an important part in expressing the oneness of the divinity. Mathematical principles expressed in number symbolisms are used to organize the world of the gods, spirits, and demons, to describe the inner structure of human beings, and to systematize mythology and theology. The concepts of duality...

symbolism in music

Number symbolism is sometimes pictorial; in the
St. Matthew Passion it is reasonable that the question “Lord, is it I?” should be asked 11 times, once by each of the faithful disciples. But the deliberate search for such symbolism in Bach’s music can be taken too far. Almost any number may be called “symbolic” (3, 6, 7, 10, 11, 12, 14, and...

writing systems

Similarly,
number systems have posed a problem for theorists because such symbols as the Arabic numerals
1,
2,
3, etc., which are conventional across many languages, appear to express thought directly without any intermediary linguistic structure. However, it is more useful to think of these numerals as a particular orthography for representing the meaning structure of...